Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.7%

Time: 12.3s

Alternatives: 14

Speedup: 1.2×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ (/ 2.0 r) r)
  (- -1.5 (/ (+ (* v -0.25) 0.375) (/ (- 1.0 v) (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) - (((v * (-0.25d0)) + 0.375d0) / ((1.0d0 - v) / ((r * w) * (r * w)))))
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / r) / r) + (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return ((2.0 / r) / r) + (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))))

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / r) / r) + Float64(-1.5 - Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 - v) / Float64(Float64(r * w) * Float64(r * w))))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / r) / r) + (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w)))));
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + N[(-1.5 - N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 88.5%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. associate-*r/ 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   5. *-commutative 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   6. associate-/l* 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   7. clear-num 89.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   8. un-div-inv 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   9. +-commutative 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. distribute-rgt-in 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. associate-*r* 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

   16. pow2 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

   17. pow2 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

   18. pow-prod-down 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
7. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]
9. Step-by-step derivation

   1. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   2. div-inv 99.7%

      \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]
11. Step-by-step derivation

    1. associate-*r/ 99.8%

       \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

    2. *-rgt-identity 99.8%

       \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

12. Simplified 99.8%

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := v \cdot -0.25 + 0.375\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -10000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + \frac{t\_0}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ (* v -0.25) 0.375)) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -10000000.0) (not (<= v 1.0)))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ v (* r w)) (* r w)))))
     (+ t_1 (+ -1.5 (/ t_0 (/ (/ (/ -1.0 r) w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (v * -0.25) + 0.375;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -10000000.0) || !(v <= 1.0)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (v * (-0.25d0)) + 0.375d0
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-10000000.0d0)) .or. (.not. (v <= 1.0d0))) then
        tmp = t_1 + ((-1.5d0) + (t_0 / ((v / (r * w)) / (r * w))))
    else
        tmp = t_1 + ((-1.5d0) + (t_0 / ((((-1.0d0) / r) / w) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (v * -0.25) + 0.375;
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -10000000.0) || !(v <= 1.0)) {
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (v * -0.25) + 0.375
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -10000000.0) or not (v <= 1.0):
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))))
	else:
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(v * -0.25) + 0.375)
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -10000000.0) || !(v <= 1.0))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(v / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(t_0 / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (v * -0.25) + 0.375;
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -10000000.0) || ~((v <= 1.0)))
		tmp = t_1 + (-1.5 + (t_0 / ((v / (r * w)) / (r * w))));
	else
		tmp = t_1 + (-1.5 + (t_0 / (((-1.0 / r) / w) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -10000000.0], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(t$95$0 / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -1e7 or 1 < v

   1. Initial program 81.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. *-commutative 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. +-commutative 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. associate-*r/ 90.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

      5. *-commutative 90.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

      6. associate-/l* 90.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

      7. clear-num 90.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      8. un-div-inv 90.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      9. +-commutative 90.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      10. distribute-rgt-in 90.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 90.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 90.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 90.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 90.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      15. associate-*r* 84.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

      16. pow2 84.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

      17. pow2 84.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

      18. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]

      2. div-inv 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \color{blue}{\left(\left(1 - v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}\right)}}\right) \]

      3. pow-flip 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)}\right) \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]

   10. Simplified 99.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]
   11. Step-by-step derivation

       1. sqr-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\left({\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)}\right)}}\right) \]

       2. pow-prod-down 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\left(\frac{-2}{2}\right)}}}\right) \]

       3. metadata-eval 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot {\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\color{blue}{-1}}}\right) \]

       4. inv-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       5. div-inv 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       6. associate-/r* 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   12. Applied egg-rr 99.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   13. Taylor expanded in v around inf 99.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{-1 \cdot \frac{v}{r \cdot w}}}{r \cdot w}}\right) \]
   14. Step-by-step derivation

       1. neg-mul-1 99.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{-\frac{v}{r \cdot w}}}{r \cdot w}}\right) \]

       2. distribute-neg-frac2 99.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{v}{-r \cdot w}}}{r \cdot w}}\right) \]

       3. distribute-rgt-neg-in 99.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\frac{v}{\color{blue}{r \cdot \left(-w\right)}}}{r \cdot w}}\right) \]

   15. Simplified 99.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{v}{r \cdot \left(-w\right)}}}{r \cdot w}}\right) \]

   if -1e7 < v < 1

   1. Initial program 87.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 87.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. *-commutative 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. +-commutative 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. associate-*r/ 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

      5. *-commutative 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

      6. associate-/l* 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

      7. clear-num 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      8. un-div-inv 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      9. +-commutative 87.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      10. distribute-rgt-in 87.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 87.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 87.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 87.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 87.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      15. associate-*r* 78.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

      16. pow2 78.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

      17. pow2 78.5%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

      18. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \color{blue}{\left(\left(1 - v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}\right)}}\right) \]

      3. pow-flip 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)}\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]
   11. Step-by-step derivation

       1. sqr-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\left({\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)}\right)}}\right) \]

       2. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\left(\frac{-2}{2}\right)}}}\right) \]

       3. metadata-eval 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot {\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\color{blue}{-1}}}\right) \]

       4. inv-pow 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       5. div-inv 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       6. associate-/r* 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   12. Applied egg-rr 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   13. Taylor expanded in v around 0 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{1}{r \cdot w}}}{r \cdot w}}\right) \]
   14. Step-by-step derivation

       1. associate-/r* 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   15. Simplified 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -10000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.9 \cdot 10^{+35} \lor \neg \left(v \leq 550000\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -2.9e+35) (not (<= v 550000.0)))
     (+ t_0 (+ -1.5 (* (* v -0.25) (* r (* (* w w) (/ r (+ v -1.0)))))))
     (+ t_0 (+ -1.5 (/ (+ (* v -0.25) 0.375) (/ (/ (/ -1.0 r) w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.9e+35) || !(v <= 550000.0)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-2.9d+35)) .or. (.not. (v <= 550000.0d0))) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = t_0 + ((-1.5d0) + (((v * (-0.25d0)) + 0.375d0) / ((((-1.0d0) / r) / w) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -2.9e+35) || !(v <= 550000.0)) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -2.9e+35) or not (v <= 550000.0):
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -2.9e+35) || !(v <= 550000.0))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -2.9e+35) || ~((v <= 550000.0)))
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = t_0 + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -2.9e+35], N[Not[LessEqual[v, 550000.0]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -2.89999999999999995e35 or 5.5e5 < v

   1. Initial program 82.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 90.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(v \cdot -0.25\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   7. Simplified 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(v \cdot -0.25\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   if -2.89999999999999995e35 < v < 5.5e5

   1. Initial program 86.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. +-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. associate-*r/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

      5. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

      6. associate-/l* 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

      7. clear-num 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      8. un-div-inv 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      9. +-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      10. distribute-rgt-in 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      15. associate-*r* 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

      16. pow2 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

      17. pow2 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

      18. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \color{blue}{\left(\left(1 - v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}\right)}}\right) \]

      3. pow-flip 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)}\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]
   11. Step-by-step derivation

       1. sqr-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\left({\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)}\right)}}\right) \]

       2. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\left(\frac{-2}{2}\right)}}}\right) \]

       3. metadata-eval 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot {\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\color{blue}{-1}}}\right) \]

       4. inv-pow 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       5. div-inv 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       6. associate-/r* 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   12. Applied egg-rr 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   13. Taylor expanded in v around 0 99.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{1}{r \cdot w}}}{r \cdot w}}\right) \]
   14. Step-by-step derivation

       1. associate-/r* 99.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   15. Simplified 99.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.9 \cdot 10^{+35} \lor \neg \left(v \leq 550000\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3100 \lor \neg \left(v \leq 550000\right):\\ \;\;\;\;t\_1 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* (* w w) (/ r (+ v -1.0))))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -3100.0) (not (<= v 550000.0)))
     (+ t_1 (+ -1.5 (* (* v -0.25) t_0)))
     (+ t_1 (+ -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3100.0) || !(v <= 550000.0)) {
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * ((w * w) * (r / (v + (-1.0d0))))
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-3100.0d0)) .or. (.not. (v <= 550000.0d0))) then
        tmp = t_1 + ((-1.5d0) + ((v * (-0.25d0)) * t_0))
    else
        tmp = t_1 + ((-1.5d0) + (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3100.0) || !(v <= 550000.0)) {
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	} else {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * ((w * w) * (r / (v + -1.0)))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -3100.0) or not (v <= 550000.0):
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0))
	else:
		tmp = t_1 + (-1.5 + (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0))))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -3100.0) || !(v <= 550000.0))
		tmp = Float64(t_1 + Float64(-1.5 + Float64(Float64(v * -0.25) * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * ((w * w) * (r / (v + -1.0)));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -3100.0) || ~((v <= 550000.0)))
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	else
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -3100.0], N[Not[LessEqual[v, 550000.0]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3100 or 5.5e5 < v

   1. Initial program 82.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 89.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative 89.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(v \cdot -0.25\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   7. Simplified 89.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(v \cdot -0.25\right)} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   if -3100 < v < 5.5e5

   1. Initial program 86.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 87.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3100 \lor \neg \left(v \leq 550000\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 7.8 \cdot 10^{+264}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.15e-86)
   (- (+ (/ (/ 2.0 r) r) 3.0) 4.5)
   (if (<= r 7.8e+264)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (+
      3.0
      (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (/ (* r w) v))) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 7.8e+264) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d-86) then
        tmp = (((2.0d0 / r) / r) + 3.0d0) - 4.5d0
    else if (r <= 7.8d+264) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * ((r * w) / v))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 7.8e+264) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15e-86:
		tmp = (((2.0 / r) / r) + 3.0) - 4.5
	elif r <= 7.8e+264:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15e-86)
		tmp = Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - 4.5);
	elseif (r <= 7.8e+264)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(Float64(r * w) / v))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15e-86)
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	elseif (r <= 7.8e+264)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * ((r * w) / v))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15e-86], N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 7.8e+264], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.14999999999999998e-86

   1. Initial program 80.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 64.1%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]
   6. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   9. Simplified 64.1%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]

   if 1.14999999999999998e-86 < r < 7.79999999999999987e264

   1. Initial program 95.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 79.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   if 7.79999999999999987e264 < r

   1. Initial program 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 87.2%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 86.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 86.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 87.2%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 93.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 93.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 93.3%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 93.3%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 99.9%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   12. Taylor expanded in v around inf 92.6%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)}\right) + 4.5\right) \]
   13. Step-by-step derivation

       1. associate-*r/ 92.6%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\frac{-1 \cdot \left(r \cdot w\right)}{v}}\right) + 4.5\right) \]

       2. neg-mul-1 92.6%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{\color{blue}{-r \cdot w}}{v}\right) + 4.5\right) \]

       3. distribute-rgt-neg-in 92.6%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{\color{blue}{r \cdot \left(-w\right)}}{v}\right) + 4.5\right) \]

   14. Simplified 92.6%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\frac{r \cdot \left(-w\right)}{v}}\right) + 4.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 7.8 \cdot 10^{+264}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 7.2 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.25e-86)
   (- (+ (/ (/ 2.0 r) r) 3.0) 4.5)
   (if (<= r 7.2e+263)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (+
      3.0
      (- (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (* r (/ w v)))) 4.5)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 7.2e+263) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (r * (w / v)))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.25d-86) then
        tmp = (((2.0d0 / r) / r) + 3.0d0) - 4.5d0
    else if (r <= 7.2d+263) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * (r * (w / v)))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.25e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 7.2e+263) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (r * (w / v)))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.25e-86:
		tmp = (((2.0 / r) / r) + 3.0) - 4.5
	elif r <= 7.2e+263:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (r * (w / v)))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.25e-86)
		tmp = Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - 4.5);
	elseif (r <= 7.2e+263)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(r * Float64(w / v)))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.25e-86)
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	elseif (r <= 7.2e+263)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (r * (w / v)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.25e-86], N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 7.2e+263], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * N[(w / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.25e-86

   1. Initial program 80.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 64.1%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]
   6. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   9. Simplified 64.1%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]

   if 1.25e-86 < r < 7.19999999999999956e263

   1. Initial program 95.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 79.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   if 7.19999999999999956e263 < r

   1. Initial program 87.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 87.2%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 86.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 86.8%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 87.2%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 93.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 93.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 93.3%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 93.3%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 93.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 99.9%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   12. Taylor expanded in v around inf 92.6%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-1 \cdot \frac{r \cdot w}{v}\right)}\right) + 4.5\right) \]
   13. Step-by-step derivation

       1. mul-1-neg 92.6%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(-\frac{r \cdot w}{v}\right)}\right) + 4.5\right) \]

       2. associate-/l* 79.1%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-\color{blue}{r \cdot \frac{w}{v}}\right)\right) + 4.5\right) \]

       3. distribute-lft-neg-in 79.1%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \frac{w}{v}\right)}\right) + 4.5\right) \]

   14. Simplified 79.1%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \frac{w}{v}\right)}\right) + 4.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.25 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 7.2 \cdot 10^{+263}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+237}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.6e-86)
   (- (+ (/ (/ 2.0 r) r) 3.0) 4.5)
   (if (<= r 3.2e+237)
     (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))
     (- 3.0 (+ 4.5 (* (* (* r w) (* r w)) (* 0.125 (+ 3.0 (* v -2.0)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.6e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 3.2e+237) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.6d-86) then
        tmp = (((2.0d0 / r) / r) + 3.0d0) - 4.5d0
    else if (r <= 3.2d+237) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (r * w)) * (0.125d0 * (3.0d0 + (v * (-2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.6e-86) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else if (r <= 3.2e+237) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.6e-86:
		tmp = (((2.0 / r) / r) + 3.0) - 4.5
	elif r <= 3.2e+237:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.6e-86)
		tmp = Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - 4.5);
	elseif (r <= 3.2e+237)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(r * w)) * Float64(0.125 * Float64(3.0 + Float64(v * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.6e-86)
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	elseif (r <= 3.2e+237)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.6e-86], N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 3.2e+237], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.6000000000000001e-86

   1. Initial program 80.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 77.0%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 64.1%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]
   6. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   7. Applied egg-rr 64.0%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   9. Simplified 64.1%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]

   if 2.6000000000000001e-86 < r < 3.20000000000000017e237

   1. Initial program 95.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 81.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{0.375} \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   if 3.20000000000000017e237 < r

   1. Initial program 89.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 89.4%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 89.1%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 89.1%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 89.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 94.5%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 94.5%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 94.5%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 94.5%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 94.5%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 94.5%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 94.5%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 99.9%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   12. Taylor expanded in v around 0 42.1%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) + 4.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{-86}:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+237}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) + 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.9e-6)
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (/ (+ (* v -0.25) 0.375) (/ (/ (/ -1.0 r) w) (* r w)))))
   (-
    3.0
    (+
     (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (/ w (/ (- 1.0 v) r))))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.9e-6) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w / ((1.0 - v) / r)))) + 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.9d-6) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (((v * (-0.25d0)) + 0.375d0) / ((((-1.0d0) / r) / w) / (r * w))))
    else
        tmp = 3.0d0 - (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * (w / ((1.0d0 - v) / r)))) + 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.9e-6) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = 3.0 - (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w / ((1.0 - v) / r)))) + 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.9e-6:
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))))
	else:
		tmp = 3.0 - (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w / ((1.0 - v) / r)))) + 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.9e-6)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(w / Float64(Float64(1.0 - v) / r)))) + 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.9e-6)
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	else
		tmp = 3.0 - (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w / ((1.0 - v) / r)))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.9e-6], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.90000000000000026e-6

   1. Initial program 81.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. *-commutative 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. +-commutative 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. associate-*r/ 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

      5. *-commutative 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

      6. associate-/l* 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

      7. clear-num 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      8. un-div-inv 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      9. +-commutative 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      10. distribute-rgt-in 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      15. associate-*r* 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

      16. pow2 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

      17. pow2 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

      18. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \color{blue}{\left(\left(1 - v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}\right)}}\right) \]

      3. pow-flip 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)}\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]
   11. Step-by-step derivation

       1. sqr-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\left({\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)}\right)}}\right) \]

       2. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\left(\frac{-2}{2}\right)}}}\right) \]

       3. metadata-eval 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot {\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\color{blue}{-1}}}\right) \]

       4. inv-pow 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       5. div-inv 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       6. associate-/r* 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   12. Applied egg-rr 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   13. Taylor expanded in v around 0 84.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{1}{r \cdot w}}}{r \cdot w}}\right) \]
   14. Step-by-step derivation

       1. associate-/r* 84.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   15. Simplified 84.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   if 5.90000000000000026e-6 < r

   1. Initial program 92.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 92.9%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 85.0%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 85.0%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 92.9%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.8%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 98.4%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   12. Step-by-step derivation

       1. clear-num 98.3%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \color{blue}{\frac{1}{\frac{1 - v}{r}}}\right)\right) + 4.5\right) \]

       2. un-div-inv 98.3%

          \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\frac{w}{\frac{1 - v}{r}}}\right) + 4.5\right) \]

   13. Applied egg-rr 98.3%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\frac{w}{\frac{1 - v}{r}}}\right) + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.9e-6)
   (+
    (/ 2.0 (* r r))
    (+ -1.5 (/ (+ (* v -0.25) 0.375) (/ (/ (/ -1.0 r) w) (* r w)))))
   (+
    3.0
    (-
     (* (* 0.125 (+ 3.0 (* v -2.0))) (* (* r w) (* w (/ r (+ v -1.0)))))
     4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.9e-6) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5.9d-6) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (((v * (-0.25d0)) + 0.375d0) / ((((-1.0d0) / r) / w) / (r * w))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + (v * (-2.0d0)))) * ((r * w) * (w * (r / (v + (-1.0d0)))))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.9e-6) {
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.9e-6:
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.9e-6)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(Float64(-1.0 / r) / w) / Float64(r * w)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.9e-6)
		tmp = (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((-1.0 / r) / w) / (r * w))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (v * -2.0))) * ((r * w) * (w * (r / (v + -1.0))))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.9e-6], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(N[(-1.0 / r), $MachinePrecision] / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.90000000000000026e-6

   1. Initial program 81.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-undefine 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      2. *-commutative 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      3. +-commutative 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

      4. associate-*r/ 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

      5. *-commutative 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

      6. associate-/l* 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

      7. clear-num 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      8. un-div-inv 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

      9. +-commutative 85.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      10. distribute-rgt-in 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      14. metadata-eval 85.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      15. associate-*r* 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

      16. pow2 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

      17. pow2 79.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

      18. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   7. Step-by-step derivation

      1. *-un-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \color{blue}{\left(\left(1 - v\right) \cdot \frac{1}{{\left(r \cdot w\right)}^{2}}\right)}}\right) \]

      3. pow-flip 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(-2\right)}}\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{\color{blue}{-2}}\right)}\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{1 \cdot \left(\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-lft-identity 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) \]
   11. Step-by-step derivation

       1. sqr-pow 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\left({\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left(r \cdot w\right)}^{\left(\frac{-2}{2}\right)}\right)}}\right) \]

       2. pow-prod-down 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\left(\frac{-2}{2}\right)}}}\right) \]

       3. metadata-eval 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot {\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}^{\color{blue}{-1}}}\right) \]

       4. inv-pow 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\left(1 - v\right) \cdot \color{blue}{\frac{1}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       5. div-inv 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

       6. associate-/r* 99.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   12. Applied egg-rr 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\color{blue}{\frac{\frac{1 - v}{r \cdot w}}{r \cdot w}}}\right) \]

   13. Taylor expanded in v around 0 84.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{1}{r \cdot w}}}{r \cdot w}}\right) \]
   14. Step-by-step derivation

       1. associate-/r* 84.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   15. Simplified 84.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{\color{blue}{\frac{\frac{1}{r}}{w}}}{r \cdot w}}\right) \]

   if 5.90000000000000026e-6 < r

   1. Initial program 92.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 92.9%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 85.0%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 85.0%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 92.9%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.8%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.8%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 98.4%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{v \cdot -0.25 + 0.375}{\frac{\frac{\frac{-1}{r}}{w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + v \cdot -2\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) + \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (- -1.5 (/ (+ (* v -0.25) 0.375) (/ (- 1.0 v) (* (* r w) (* r w)))))
  (/ 2.0 (* r r))))

C

double code(double v, double w, double r) {
	return (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w))))) + (2.0 / (r * r));
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((-1.5d0) - (((v * (-0.25d0)) + 0.375d0) / ((1.0d0 - v) / ((r * w) * (r * w))))) + (2.0d0 / (r * r))
end function

Java

public static double code(double v, double w, double r) {
	return (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w))))) + (2.0 / (r * r));
}

Python

def code(v, w, r):
	return (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w))))) + (2.0 / (r * r))

Julia

function code(v, w, r)
	return Float64(Float64(-1.5 - Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(Float64(1.0 - v) / Float64(Float64(r * w) * Float64(r * w))))) + Float64(2.0 / Float64(r * r)))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (-1.5 - (((v * -0.25) + 0.375) / ((1.0 - v) / ((r * w) * (r * w))))) + (2.0 / (r * r));
end

Wolfram

code[v_, w_, r_] := N[(N[(-1.5 - N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 88.5%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 88.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. associate-*r/ 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   5. *-commutative 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   6. associate-/l* 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   7. clear-num 89.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   8. un-div-inv 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   9. +-commutative 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. distribute-rgt-in 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 88.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + \color{blue}{0.375}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. associate-*r* 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot w\right)}}}\right) \]

   16. pow2 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{r}^{2}} \cdot \left(w \cdot w\right)}}\right) \]

   17. pow2 81.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{r}^{2} \cdot \color{blue}{{w}^{2}}}}\right) \]

   18. pow-prod-down 99.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{{\left(r \cdot w\right)}^{2}}}}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{{\left(r \cdot w\right)}^{2}}}}\right) \]
7. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}}\right) \]

9. Final simplification 99.8%

   \[\leadsto \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) + \frac{2}{r \cdot r} \]
10. Add Preprocessing

Alternative 11: 68.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5800000:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5800000.0)
   (- (+ (/ (/ 2.0 r) r) 3.0) 4.5)
   (- 3.0 (+ 4.5 (* (* (* r w) (* r w)) (* 0.125 (+ 3.0 (* v -2.0))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5800000.0) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5800000.0d0) then
        tmp = (((2.0d0 / r) / r) + 3.0d0) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (r * w)) * (0.125d0 * (3.0d0 + (v * (-2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5800000.0) {
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5800000.0:
		tmp = (((2.0 / r) / r) + 3.0) - 4.5
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5800000.0)
		tmp = Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(r * w)) * Float64(0.125 * Float64(3.0 + Float64(v * -2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5800000.0)
		tmp = (((2.0 / r) / r) + 3.0) - 4.5;
	else
		tmp = 3.0 - (4.5 + (((r * w) * (r * w)) * (0.125 * (3.0 + (v * -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5800000.0], N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.8e6

   1. Initial program 82.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   3. Simplified 78.5%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 64.4%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]
   6. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.7%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   7. Applied egg-rr 64.3%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   9. Simplified 64.4%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]

   if 5.8e6 < r

   1. Initial program 92.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Step-by-step derivation

      1. associate--l- 92.5%

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]

      2. associate-*l* 84.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}{1 - v} + 4.5\right) \]

      3. sqr-neg 84.3%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot w\right) \cdot \color{blue}{\left(\left(-r\right) \cdot \left(-r\right)\right)}\right)}{1 - v} + 4.5\right) \]

      4. associate-*l* 92.5%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)\right)}}{1 - v} + 4.5\right) \]

      5. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}} + 4.5\right) \]

      6. fma-define 98.4%

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(0.125 \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot \left(-r\right)\right) \cdot \left(-r\right)}{1 - v}, 4.5\right)} \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   6. Applied egg-rr 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   7. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   8. Simplified 98.4%

      \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v} + 4.5\right) \]
   9. Step-by-step derivation

      1. associate-/l* 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(r \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right)} + 4.5\right) \]

      2. *-commutative 98.4%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{\left(w \cdot w\right) \cdot r}}{1 - v}\right) + 4.5\right) \]

      3. associate-*r/ 98.3%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)}\right) + 4.5\right) \]

      4. associate-*l* 99.8%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)}\right) + 4.5\right) \]

      5. associate-*r* 99.9%

         \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   10. Applied egg-rr 99.9%

       \[\leadsto \left(3 + \frac{\frac{2}{r}}{r}\right) - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)} + 4.5\right) \]

   11. Taylor expanded in r around inf 99.9%

       \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]

   12. Taylor expanded in v around 0 64.6%

       \[\leadsto 3 - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5800000:\\ \;\;\;\;\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(0.125 \cdot \left(3 + v \cdot -2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- (+ (/ (/ 2.0 r) r) 3.0) 4.5))

C

double code(double v, double w, double r) {
	return (((2.0 / r) / r) + 3.0) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (((2.0d0 / r) / r) + 3.0d0) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return (((2.0 / r) / r) + 3.0) - 4.5;
}

Python

def code(v, w, r):
	return (((2.0 / r) / r) + 3.0) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(Float64(2.0 / r) / r) + 3.0) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (((2.0 / r) / r) + 3.0) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 79.1%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 55.4%

   \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]
6. Step-by-step derivation

   1. associate-/r* 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   2. div-inv 99.7%

      \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

7. Applied egg-rr 55.3%

   \[\leadsto \left(3 + \color{blue}{\frac{2}{r} \cdot \frac{1}{r}}\right) - 4.5 \]
8. Step-by-step derivation

   1. associate-*r/ 99.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{r} \cdot 1}{r}} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

   2. *-rgt-identity 99.8%

      \[\leadsto \frac{\color{blue}{\frac{2}{r}}}{r} + \left(-1.5 - \frac{v \cdot -0.25 + 0.375}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) \]

9. Simplified 55.4%

   \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]

10. Final simplification 55.4%

    \[\leadsto \left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5 \]
11. Add Preprocessing

Alternative 13: 57.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{2}{r \cdot r} + 3\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (- (+ (/ 2.0 (* r r)) 3.0) 4.5))

C

double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((2.0 / (r * r)) + 3.0) - 4.5;
}

Python

def code(v, w, r):
	return ((2.0 / (r * r)) + 3.0) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 79.1%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 55.4%

   \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]

6. Final simplification 55.4%

   \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - 4.5 \]
7. Add Preprocessing

Alternative 14: 14.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.6%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

3. Simplified 79.1%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 55.4%

   \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{4.5} \]

6. Taylor expanded in r around inf 14.3%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))